In recent years, a remarkably large number of inequalities involving the fractional q-integral operators have been investigated in the literature by many authors. Here, we aim to present some new fractional integral inequalities involving generalized Erdélyi-Kober fractional q-integral operator due to Gaulué, whose special cases are shown to yield corresponding inequalities associated with Kober type fractional q-integral operators. The cases of synchronous functions as well as of functions bounded by integrable functions are considered.

The inequality in (2) is reversed if f and g are asynchronous on [a, b]; that is,

[Formula ID: EEq1.4](4)

(f(x)−f(y))(g(x)−g(y))≤0,

for any x, y ∈ [a, b]. If p(x) = q(x) for any x, y ∈ [a, b], we get the Chebyshev inequality (see [1]). Ostrowski [4] established the following generalization of the Chebyshev inequality.

If f and g are two differentiable and synchronous functions on [a, b] and p is a positive integrable function on [a, b] with |f′(x)|≥m and |g′(x)|≥r for x ∈ [a, b], then we have

[Formula ID: EEq1.5](5)

T(f,g,p)=T(f,g,p,p)≥mrT(x−a,x−a,p)≥0.

If f and g are asynchronous on [a, b], then we have

[Formula ID: EEq1.6](6)

T(f,g,p)≤mrT(x−a,x−a,p)≤0.

If f and g are two differentiable functions on [a, b] with |f′(x)|≤M and |g′(x)|≤R for x ∈ [a, b] and p is a positive integrable function on [a, b], then we have

[Formula ID: EEq1.7](7)

|T(f,g,p)|≤MRT(x−a,x−a,p)≤0.

Here, it is worth mentioning that the functional (1) has attracted many researchers' attention mainly due to diverse applications in numerical quadrature, transform theory, probability, and statistical problems. Among those applications, the functional (1) has also been employed to yield a number of integral inequalities (see, e.g., [5–11]).

The study of the fractional integral and fractional q-integral inequalities has been of great importance due to the fundamental role in the theory of differential equations. In recent years, a number of researchers have done deep study, that is, the properties, applications, and different extensions of various fractional q-integral operators (see, e.g., [12–16]).

The purpose of this paper is to find q-calculus analogs of some classical integral inequalities. In particular, we will find q-generalizations of the Chebyshev integral inequalities by using the generalized Erdélyi-Kober fractional q-integral operator introduced by Galué [17]. The main objective of this paper is to present some new fractional q-integral inequalities involving the generalized Erdélyi-Kober fractional q-integral operator. We consider the case of synchronous functions as well as the case of functions bounded by integrable functions. Some of the known and new results are as follows, as special cases of our main findings. We emphasize that the results derived in this paper are more generalized results rather than similar published results because we established all results by using the generalized Erdélyi-Kober fractional q-integral operator. Our results are general in character and give some contributions to the theory q-integral inequalities and fractional calculus.

2. Preliminaries

In the sequel, we required the following well-known results to establish our main results in the present paper. The q-shifted factorial (a; q)n is defined by

for all μ > 0 and k ∈ N0. If f : [0, ∞)→[0, ∞) is a continuous function, then we conclude that, under the given conditions in (26), each term in the series of generalized Erdélyi-Kober q-integral operator is nonnegative and thus

[Formula ID: EEq2.20](29)

Iqη,μ,β{f}(t)≥0,

for all β, μ > 0 and η ∈ C.

On the same way each term in the series of Kober q-integral operator (27) is also nonnegative and thus

Next, multiplying both sides of (34) by (βt−β(η+μ)/Γq(μ))(tβ − ρβq)(μ−1)ρβ(η+1)−1v(ρ), integrating the resulting inequality with respect to ρ from 0 to t, and using (26), we are led to the desired result (31).

Similarly replacing u, v by l, n and u, v by l, m, respectively, in (31) and then multiplying both sides of the resulting inequalities by Iqη,μ,β{m}(t) and Iqη,μ,β{n}(t) both of which are nonnegative under the given assumptions, respectively, we get the following inequalities:

Now, by replacing u, v by l, n and u, v by l, m in (35), respectively, and then multiplying both sides of the resulting inequalities by Iqη,μ,β{m}(t) and Iqη,μ,β{n}(t), respectively, we get the following two inequalities:

Finally, we find that the inequality (42) follows by adding the inequalities (44) and (45), side by side.

Remark 8 .

It may be noted that inequalities (37) and (42) in Theorems 6 and 7, respectively, are reversed if the functions are asynchronous on [0, ∞). The special case of (42) in Theorem 7 when β = δ, η = ζ, and μ = ν is easily seen to yield inequality (37) in Theorem 6.

Remark 9 .

We remark further that we can present a large number of special cases of our main inequalities in Theorems 6 and 7. Here, we give only two examples: setting β = 1 in (37) and β = δ = 1 in (42), we obtain interesting inequalities involving Erdélyi-Kober fractional integral operator.

In this section we obtain some new inequalities involving Erdélyi-Kober fractional q-integral operator in the case where the functions are bounded by integrable functions and are not necessary increasing or decreasing as are the synchronous functions.

Multiplying both sides of (52) by (βt−β(η+μ)/Γq(μ))(tβ − ρβq)(μ−1)ρβ(η+1)−1v(ρ), ρ ∈ (0, t), and integrating both sides with respect to ρ on (0, t), we get inequality (49) as requested. This completes the proof.

for any k > 0. Multiplying both sides of (69) by (βt−β(η+μ)/Γq(μ))(tβ − τβq)(μ−1)τβ(η+1)−1u(τ), τ ∈ (0, t), and integrating the resulting identity with respect to τ from 0 to t, one has inequality (i). Inequality (ii) is proved by setting a = f(τ) − φ1(τ) in Lemma 20.

We conclude our present investigation with the remark that the results derived in this paper are general in character and give some contributions to the theory of q-integral inequalities and fractional calculus. Moreover, they are expected to find some applications for establishing uniqueness of solutions in fractional boundary value problems and in fractional partial differential equations. In last, use of the generalized Erdélyi-Kober fractional q-integral operator due to Gaulué is the advantage of our results because after setting suitable parameter values in our main results, we get known results established by number of authors.

Acknowledgments

The research of J. Tariboon is supported by King Mongkut's University of Technology North Bangkok, Thailand. Sotiris K. Ntouyas is a Member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.